where the set of “quasi-momenta" \({p_i}\) , no two of which are identical [1]}, will be determined by the periodicity conditions. \(\mathcal {P}\)  represents permutations of the quasi-momenta and \(\mathcal {Q}\)  is the particle order: the permutation satisfying \(x_{\mathcal {Q}^{-1}(i+1)} > x_{\mathcal {Q}^{-1}(i)}\) . Amplitudes are permutation invariant \(\mathcal {A}_\mathcal {P}(\mathcal {Q})= \mathcal {A}_{\mathcal {P}\circ \mathcal {P}^{\prime }}(\mathcal {Q}\circ \mathcal {P}^{\prime })\)  for all \(\mathcal {P}^{\prime }\) , which guarantees permutation invariance of the bosonic wave-function. Amplitudes are fixed by the delta-function interaction, and their ratios give rise to scattering phases, as the scattering problem can be obtained from the limit \(L\rightarrow \infty \) . With \(\mathcal {P^{\prime }}\)  differing from \(\mathcal {P}\)  only through permutation of quasi-particles \(i\)  and \(j\) , the ratio involves the two-particle phase shift \(\phi (p_i,p_j)\) ,
\(\frac{\mathcal {A}_\mathcal {P}}{\mathcal {A}_\mathcal {P ^{\prime }}} =\exp \left[{\rm i}\phi (p_i,p_j)\right] =\frac{p_i-p_j -{\rm i}c}{p_i-p_j +{\rm i}c} \; .\) 
